Question

let abc be a triangle such that angle a is opposite side a, angle b is opposite side b, and angle c is opposite side c. if ( b = 112.26^{circ} ), ( a = 13 ), and ( b = 15 ), solve for angle a. round your answer to the nearest hundredth of a degree.
( mangle a=)

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}\).
So, \(\sin A=\frac{a\sin B}{b}\).

Step2: Substitute the given values

Given \(a = 13\), \(b = 15\), and \(B=112.26^{\circ}\).
First, \(\sin B=\sin(112.26^{\circ})\approx0.925\).
Then \(\sin A=\frac{13\times0.925}{15}\).
\(\sin A=\frac{12.025}{15}\approx0.8017\).

Step3: Find angle \(A\)

\(A=\sin^{- 1}(0.8017)\).
Using a calculator, \(A\approx53.21^{\circ}\).

Answer:

\(53.21\)